1 Dynamic Behavior Chapter 5 In analyzing process dynamic and process control systems, it is important to know how the process responds to changes in the process inputs. A number of standard types of input changes are widely used for two reasons: 1.They are representative of the types of changes that occur in plants. 2.They are easy to analyze mathematically.

2 1.Step Input A sudden change in a process variable can be approximated by a step change of magnitude, M: Chapter 5 Special Case: If M = 1, we have a “unit step change”. We give it the symbol, S(t). Example of a step change: A reactor feedstock is suddenly switched from one supply to another, causing sudden changes in feed concentration, flow, etc. The step change occurs at an arbitrary time denoted as t = 0.

3 Chapter 5 Example: The heat input to the stirred-tank heating system in Chapter 2 is suddenly changed from 8000 to 10,000 kcal/hr by changing the electrical signal to the heater. Thus, and 2.Ramp Input Industrial processes often experience “drifting disturbances”, that is, relatively slow changes up or down for some period of time. The rate of change is approximately constant.

4 Chapter 5 We can approximate a drifting disturbance by a ramp input: Examples of ramp changes: 1.Ramp a setpoint to a new value. (Why not make a step change?) 2.Feed composition, heat exchanger fouling, catalyst activity, ambient temperature. 3.Rectangular Pulse It represents a brief, sudden change in a process variable:

5 Chapter 5 Examples: 1.Reactor feed is shut off for one hour. 2.The fuel gas supply to a furnace is briefly interrupted. 0 h X RP T w Time, t

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7 Examples: 1.24 hour variations in cooling water temperature Hz electrical noise (in the USA) 4.Sinusoidal Input Processes are also subject to periodic, or cyclic, disturbances. They can be approximated by a sinusoidal disturbance: where: A = amplitude,  = angular frequency

8 Chapter 5 Examples: 1.Electrical noise spike in a thermo-couple reading. 2.Injection of a tracer dye. 5.Impulse Input Here, It represents a short, transient disturbance. Useful for analysis since the response to an impulse input is the inverse of the TF. Thus, Here,

9 Chapter 5 The corresponding time domain express is: where: Suppose. Then it can be shown that: Consequently, g(t) is called the “impulse response function”.

10 Chapter 5 The standard form for a first-order TF is: where: Consider the response of this system to a step of magnitude, M: Substitute into (5-16) and rearrange, First-Order System

11 Chapter 5 Take L -1 (cf. Table 3.1), Let steady-state value of y(t). From (5-18), t___ Note: Large means a slow response.

12 Chapter 5 Consider a step change of magnitude M. Then U(s) = M/s and, Integrating Process Not all processes have a steady-state gain. For example, an “integrating process” or “integrator” has the transfer function: Thus, y(t) is unbounded and a new steady-state value does not exist. L -1

13 Chapter 5 Consider a liquid storage tank with a pump on the exit line: Common Physical Example: -Assume: 1.Constant cross-sectional area, A. 2. -Mass balance: -Eq. (1) – Eq. (2), take L, assume steady state initially, -For (constant q),

14 Chapter 5 Standard form: Second-Order Systems which has three model parameters: Equivalent form:

15 Chapter 5 The type of behavior that occurs depends on the numerical value of damping coefficient, : It is convenient to consider three types of behavior: Damping Coefficient Type of ResponseRoots of Charact. Polynomial OverdampedReal and ≠ Critically dampedReal and = UnderdampedComplex conjugates Note: The characteristic polynomial is the denominator of the transfer function: What about ? It results in an unstable system

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18 Chapter 5 1.Responses exhibiting oscillation and overshoot (y/KM > 1) are obtained only for values of less than one. 2.Large values of yield a sluggish (slow) response. 3.The fastest response without overshoot is obtained for the critically damped case Several general remarks can be made concerning the responses show in Figs. 5.8 and 5.9:

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20 Chapter 5 1.Rise Time: is the time the process output takes to first reach the new steady-state value. 2.Time to First Peak: is the time required for the output to reach its first maximum value. 3.Settling Time: is defined as the time required for the process output to reach and remain inside a band whose width is equal to ±5% of the total change in y. The term 95% response time sometimes is used to refer to this case. Also, values of ±1% sometimes are used. 4.Overshoot: OS = a/b (% overshoot is 100a/b). 5.Decay Ratio: DR = c/a (where c is the height of the second peak). 6.Period of Oscillation: P is the time between two successive peaks or two successive valleys of the response.